# On Trivial p-adic Zeroes for Elliptic Curves Over Kummer Extensions

### New Zealand Journal of Mathematics

Vol. 45, (2015), Pages 33-38

Daniel Delbourgo

Department of Mathematics,

University of Waikato,

Private Bag 3105, Hamilton,

New Zealand

Abstract We prove the exceptional zero conjecture is true for semistable elliptic curves $E_{/\mathbb{Q}}$ over number fields of the form $F\big(e^{2\pi i/q^n},\Delta_1^{1/q^n},\dots,\Delta_d^{1/q^n}\big)$ where F is a totally real field, and the split multiplicative prime $p\neq 2$ is inert in $F(e^{2\pi i/q^n})\cap \R$.

Keywords p-adic L-functions, elliptic curves, Mazur-Tate-Teitelbaum conjecture.

Classification (MSC2000) 11F33, 11F41, 11F67.